## Abstract

Techniques from electrical network theory have been used to establish various properties of random walks. We explore this connection further, by showing how the classical formulas for the determinant and cofactors of the admittance matrix, due to Maxwell and Kirchoff, yield upper bounds on the edge stretch factor of the harmonic random walk. For any complete, n-vertex graph with distances assigned to its edges, we show the upper bound of (n - 1)^{2}. If the distance function satisfies the triangle inequality, we give the upper bound of 12n(n - 1). Both bounds are tight. As a consequence, we obtain that the harmonic algorithm for the k server problem is 12k(k + 1)-competitive against the lazy adversary.

Original language | American English |
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Pages (from-to) | 271-276 |

Number of pages | 6 |

Journal | Information Processing Letters |

Volume | 84 |

Issue number | 5 |

DOIs | |

State | Published - 16 Dec 2002 |

### Bibliographical note

Funding Information:* Corresponding author. Research supported by NSF grant CCR-9988360 and cooperative grant KONTAKT-ME476/CCR-9988360-001 from MŠMT Cˇ R and NSF. E-mail address: marek@cs.ucr.edu (M. Chrobak). 1 Research supported by NSF grant CCR-9988360.

## Keywords

- Analysis of algorithms
- Electrical networks
- Randomization
- Theory of computation